Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Spinors
SpinorMultivector | |
PauliSpinor | |
DiracSpinor |
Pauli
Pauli
In[77]:=
Needs["Wolfram`GeometricAlgebra`PauliDirac`"]
In[51]:=
$PauliAlgebra["Basis",{0,1,-1}]
Out[51]=
In[39]:=
{,,,,ℐ}=$PauliAlgebra["Basis",{0,1,-1}];{,,}=$PauliAlgebra["PseudoBasis",1];
σ
1
σ
2
σ
3
iσ
1
iσ
2
iσ
3
In[105]:=
ϕ=SpinorMultivector[Table[Superscript[a,i],{i,0,3}]]η=SpinorMultivector[Table[Superscript[b,i],{i,0,3}]]
Out[105]=
0
a
3
a
σ
12
2
a
σ
13
1
a
σ
23
Out[106]=
0
b
3
b
σ
12
2
b
σ
13
1
b
σ
23
In[56]:=
PauliSpinor[ϕ]
Out[56]=
0 a 3 a |
- 2 a 1 a |
In[114]:=
SpinorMultivector@PauliSpinor[ϕ]ϕ
Out[114]=
True
Operating with Pauli matrix on multivector spinor |ϕ〉ϕ:
σ
i
σ
i
σ
3
In[107]:=
Table[SpinorMultivector[PauliMatrix[i].PauliSpinor[ϕ]]**ϕ**,{i,3}]
σ
i
σ
3
Out[107]=
{True,True,True}
Imaginary unit multiplication :
i|ϕ〉ϕℐ
σ
3
In[108]:=
SpinorMultivector[IPauliSpinor[ϕ]]ϕ**ℐ**ϕ**
σ
3
iσ
3
Out[108]=
True
In $PauliSpectralBasis basis vectors has an (almost) familiar form of Pauli matrices:
In[52]:=
MatrixForm@Normal@MultivectorMatrix[#,"Basis"$PauliSpectralBasis]&/@{,,}
σ
1
σ
2
σ
3
Out[52]=
In[176]:=
PauliMatrix[2]
Out[176]=
0 | - |
| 0 |
Second Pauli matrix looks like it has a wrong sign, but in the isomorphism ≡ pseudoscalar of of matrix components represents pseudoscalar of the “complex” algebra for or depending on a commutativity of the pseudoscalar), which is .But pseudoscalar converted from to acquires a minus sign:
3
M
2×2
0,1
0,1
3
n+1,n
n,n+1
1,2
1,2
3
In[180]:=
$PauliAlgebra["ComplexAlgebra"]
Out[180]=
1,2
In[179]:=
 | ConvertGeometricAlgebra |
Out[179]=
-ℐ
A transformation that takes this into account is ComplexMultivector, that leaves only scalar and pseudoscalar parts with the correct multiplier:
In[187]:=
MultivectorMatrix[,"Basis"$PauliSpectralBasis][ComplexMultivector[#,$PauliAlgebra]&]
σ
2
Out[187]=
Converting directly from a familiar Pauli matrix to :
3
In[53]:=
Table[MatrixMultivector[MultivectorArray[PauliMatrix[i]],$PauliAlgebra,"Basis"$PauliSpectralBasis],{i,0,3}]
Out[53]=
Again second Pauli matrix does not directly converts to , but imaginary unit in this case is supposed to represent pseudoscalar of .
A conversion of imaginary part in multivector coordinates as though it is a pseudoscalar (on the left) can be done with “Real” property”
σ
2
3
A conversion of imaginary part in multivector coordinates as though it is a pseudoscalar (on the left) can be done with “Real” property”
In[58]:=
Map[#["Real"]&,Table[MatrixMultivector[MultivectorArray[PauliMatrix[i]],$PauliAlgebra,"Basis"$PauliSpectralBasis],{i,3}]]$PauliBasis
Out[58]=
True
Dirac
Dirac
Dirac matrices from vectors of :
i
γ
1,3
In[87]:=
{,,,}=$DiracAlgebra["Basis",1];
γ
0
γ
1
γ
2
γ
3
In[85]:=
Table[MatrixForm@Normal@MultivectorMatrix[,"Basis"$DiracSpectralBasis],{i,0,3}]
γ
i
Out[85]=
And the covariant version :
γ
i
In[86]:=
Table[MatrixForm@Normal@MultivectorMatrix[,"Basis"$DiracCovariantSpectralBasis],{i,0,3}]
γ
i
Out[86]=
Backwards from Dirac matrices to multivectors:
In[88]:=
Table[MatrixMultivector[MultivectorArray[DiracMatrix[i]],$DiracAlgebra,"Basis"$DiracSpectralBasis],{i,0,3}]
Out[88]=
In[89]:=
Table[MatrixMultivector[MultivectorArray[DiracMatrix[i]],$DiracAlgebra,"Basis"$DiracCovariantSpectralBasis],{i,0,3}]
Out[89]=
In[91]:=
Inverse/@{,,,}
γ
0
γ
1
γ
2
γ
3
Out[91]=
Dirac spinor:
In[109]:=
ψ=SpinorMultivector[Table[{Superscript[φ,i]},{i,0,3}]]
Out[109]=
-++-++-+-++-+-++-++-++-+-
0
φ
4
0
φ
4
γ
0
3
φ
4
γ
1
3
φ
4
γ
2
2
φ
4
γ
3
3
φ
4
γ
01
3
φ
4
γ
02
2
φ
4
γ
03
0
φ
4
γ
12
1
φ
4
γ
13
1
φ
4
γ
23
0
φ
4
γ
012
1
φ
4
γ
013
1
φ
4
γ
023
2
φ
4
γ
123
2
φ
4
γ
5
In[95]:=
Normal@MultivectorMatrix[ψ,"Basis"$DiracSpectralBasis]
Out[95]=
In[94]:=
DiracSpinor[ψ]
Out[94]=
0 φ |
1 φ |
2 φ |
3 φ |
Multiplying spinor by Dirac gamma matrix |ψ〉ψ
γ
i
γ
i
γ
0
In[110]:=
Table[SpinorMultivector[DiracMatrix[i].DiracSpinor[ψ]]**ψ**,{i,0,3}]
γ
i
γ
0
Out[110]=
{True,True,True,True}
Multiplying by imaginary unit
|ψ〉ψℐ=-ψ
σ
3
γ
12
In[111]:=
Table[SpinorMultivector[DiracSpinor[ψ]]-ψ**(**),{i,0,3}]
γ
1
γ
2
Out[111]=
{True,True,True,True}
""